On the acoustic analysis and optimization of ducted ventilationsystems using a sub-structuring approach

X. Yu,1 F. S. Cui,1 and L. Cheng2,a)

1Institute of High Performance Computing, Agency for Science, Technology and Research, Singapore, Singapore2Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China

(Received 18 November 2015; revised 16 December 2015; accepted 28 December 2015; publishedonline 13 January 2016)

This paper presents a general sub-structuring approach to predict the acoustic performance of

ducted ventilation systems. The modeling principle is to determine the subsystem characteristics by

calculating the transfer functions at their coupling interfaces, and the assembly is enabled by using

a patch-based interface matching technique. For a particular example of a bended ventilation duct

connecting an inlet and an outlet acoustic domain, a numerical model is developed to predict its

sound attenuation performance. The prediction accuracy is thoroughly validated against finite ele-

ment models. Through numerical examples, the rigid-walled duct is shown to provide relatively

weak transmission loss (TL) across the frequency range of interest, and exhibit only the reactive

behavior for sound reflection. By integrating sound absorbing treatment such as micro-perforated

absorbers into the system, the TL can be significantly improved, and the system is seen to exhibit

hybrid mechanisms for sound attenuation. The dissipative effect dominates at frequencies where

the absorber is designed to be effective, and the reactive effect provides compensations at the

absorption valleys attributed to the resonant behavior of the absorber. This ultimately maintains the

system TL at a relatively high level across the entire frequency of interest. The TL of the system

can be tuned or optimized in a very efficient way using the proposed approach due to its modular

nature. It is shown that a balance of the hybrid mechanism is important to achieve an overall broad-

band attenuation performance in the design frequency range.VC 2016 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4939785]

[NX] Pages: 279–289

I. INTRODUCTION

Natural ventilation in buildings is an attractive topic for

living comfort improvement and sustainable development.

The airflow and thermal performances as the primary factors

have been extensively studied.1 However, ventilation aper-

tures by opening large areas of building facades offer little

resistance to noise passage. This could limit the potential of

using natural ventilation in high noise concentration areas

such as the major metropolitan cities. Typically, for an

acoustically hard-walled building facade containing an aper-

ture, its sound reduction index tends to be dominated by the

aperture, even though its area is much smaller than that of

the wall.2

There always seems to be a conflict between natural

ventilation and acoustic insulation. Among possible acoustic

treatments reported in the literature, the sound barrier2 and

acoustic louver3 are effective in screening the direct sound

incidence in relatively high frequency range, but they cannot

handle low frequency noise due to the long wavelength char-

acter of sound. The quarter wave or Helmholtz resonators4

can be specially designed for suppressing the low frequency

noise within a narrow frequency band, but the attenuation

provided for an aperture is usually weak due to the limited

space available.

The strategy of using a ducted ventilation system may

provide possibilities to reduce the noise ingress while main-

taining sufficient airflow passage. In general, the noise con-

trol treatment has more opportunity to be applied inside the

ductwork compared with the limited space surrounding an

aperture. For example, De Salis et al.5 reviewed a number of

noise control techniques for naturally ventilated buildings,

and suggested that a ventilation duct could embrace the use

of absorbing treatments as well as active noise cancellation.

Kang and Brocklesby6 proposed a ventilation window design

by staggering two layers of glass panels, between which

transparent micro-perforated absorbers were added to reduce

the sound transmission. Numerical simulations were later

conducted by varying window design parameters,7 and

Huang et al.8 further applied an active noise control tech-

nique to suppress the low frequency noise penetrating

through the window system. Moreover, Wang et al.9 com-

pared the effectiveness of using quarter wave resonators or

membrane absorbers in a ducted ventilation window.

Concurrent use of both active noise and vibration control in

a ventilation duct was also examined by Rohlfing and

Gardonio.10

Whilst the potential of using a ducted ventilation system

for noise control is promising, research into the numerical

analysis and optimization of its acoustic performance has

been fairly limited, possibly due to the numerous challenges

faced by the conventional modeling approaches. For exam-

ple, analytical approaches based on the modal description ofa)Electronic mail: li.cheng@polyu.edu.hk

J. Acoust. Soc. Am. 139 (1), January 2016 VC 2016 Acoustical Society of America 2790001-4966/2016/139(1)/279/11/$30.00

acoustic waves can only handle simple system geometries.

The Finite Element Method (FEM) provides a powerful tool

in solving coupled physical problems, but the number of dis-

cretized nodes and induced computational cost drastically

increase as the system dimension is large or the targeted fre-

quencies are high. This becomes even more crucial when

parametric studies or system optimizations are needed. In

contrast to the deterministic models, Statistical Energy

Analysis offers a rough analysis tool at high frequency

range.11 But it may not be applicable here since the noise

ingress into the building mainly concentrates in the low-to-

mid frequency range. Also, the results predicted from a

statistical approach are too rough to enable a clear under-

standing of the sound transmission mechanisms.

In tackling these challenges, this paper presents a general

sub-structuring approach for the acoustic analysis and optimi-

zation of ducted ventilation systems. The basic principle is to

decouple the complex system into a number of subsystems

which can be modeled separately, and the overall system

response is solved by using a transfer function based assem-

bling treatment. For a convenient interface description, the

connecting surface between two subsystems is meshed into a

set of patch elements. It is assumed that the distributed points

within a patch exhibit a similar vibrational response under ex-

citation, so that an averaged property is enough to characterize

the behavior of each patch. The validity of this assumption

can be ensured by the patch division criteria compared with

the minimum acoustic wavelength of interest.12 In this way,

the patch based interface matching requires less number of

degrees of freedom (n-DOF) compared with the traditional

point collocation technique,13 and the required computational

cost is drastically reduced. Meanwhile, the determination of

the subsystem properties using the proposed approach can be

performed by using an open selection of numerical or even

experimental tools.12 This provides a more flexible option for

subsystem description compared with the analytical mode

matching technique.14

In this paper, the modeling framework of the proposed

sub-structuring approach is first established by considering a

generic ducted ventilation system in Sec. II, where the general

structural-acoustic coupling using the patch based interface

matching is formulated. A numerical model is further devel-

oped for the prediction of its sound attenuation performance.

Given a particular system configuration in Sec. III, the predic-

tion accuracy using the proposed approach is validated against

FEM simulations, and the associated computational costs are

compared. Then, non-fibrous micro-perforated panel (MPP)

absorbers are used inside the ventilation duct to increase its

sound attenuation performance. The study not only demon-

strates the effectiveness of MPP through sound absorption

enhancement, but also reveals the underlying sound attenua-

tion mechanisms. Based on a brief parametric study, the sensi-

tivity of the system response to parameter variations is

discussed, suggesting that a tunable acoustic performance can

be achieved in a targeted frequency range. Subsequently, two

optimization case studies are presented by tuning the associ-

ated parameters, which demonstrates the capability and the

computation efficiency of the proposed approach.

II. FORMULATION

Consider a generic example representing a ducted venti-

lation system being inserted into a building facade as shown

in Fig. 1. From the acoustic viewpoint, the ventilation duct is

simply an acoustic waveguide connecting the exterior and

interior acoustic media. The ductwork incorporates spatial

distortions such as bends to avoid significant length, which

also helps screen the direct path for noise ingress. Without

loss of generality, the ventilation duct in Fig. 1 can be either

rigid-walled or integrated with noise control treatments such

as porous absorbent material and non-fibrous MPP absorb-

ers. The incident sound waves reaching the inlet aperture can

be due to any sound sources such as road traffic, and the

transmitted sound waves from the outlet aperture describe

the noise leakage into the building interior.

The entire system is divided into three acoustic subsys-

tems and two coupling interfaces, I and II, as shown in Fig. 1.

Each interface may be defined implicitly as a vibrating bound-

ary surface like a flexible panel. For convenience in the cou-

pling treatment, each interface is segmented into a set of

patches, acting as the coupling agents or energy channels

between subsystems. The two interfaces, I and II, in Fig. 1 are

meshed into N and N0 patches, respectively. For deterministic

analysis, the patch size needs to be properly selected, small

enough to describe the short-wavelength behavior of the

waves. An appropriate patch division criteria of Dpatch < ks=2

has been suggested and discussed by Ouisse et al. in a previ-

ous publication,12 where Dpatch is the required patch size and

ks is the minimum acoustic wavelength of interest.

For a general separation interface with N patches seg-

mented over its surface, the relationship between the excita-

tion force and velocity response can be written as

U ¼ Y � F5

Y11 Y12 � � �

Y21. .

.

..

.YNN

2664

3775 �

F1

..

.

FN

2664

3775 ¼

U1

..

.

UN

2664

3775;

(1)

where Y is the mobility matrix of the interface, a square ma-

trix of order N; F is the column vector of the excitation force

applied to each patch, both the mechanical and acoustical

FIG. 1. An example of a ducted ventilation system being inserted into a

building facade.

280 J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al.

excitations are combined; and U is the column vector of the

averaged patch velocity, as the steady-state response of the

interface.

If the separation boundary presents itself in a fully

coupled vibroacoustic environment, its vibration will also

cause disturbance to the connected acoustic domains. The

acoustic loading on either side of the interface due to such

coupling can be expressed as

Fa ¼ Za � U; (2)

where Za is the impedance matrix of the acoustic domain

and Fa is the acoustic loading force resulted from the vibra-

tion of the interface.

Then, based on the linear superposition principle, the

generalized structural-acoustic coupling formulation for an

interface can be described as

Y � ðFext þ Zl � Uþ Zr � UÞ ¼ U; (3)

where Zl and Zr are the impedances of the coupled acoustic

domain, e.g., on the left- and right-hand side of the interface.

Fext describes any external excitation force applied to the

interface.

In special cases when the interface is better character-

ized as an impedance surface, Eq. (3) changes to

Fext þ Zl � Uþ Zr � U ¼ Z � U; (4)

where Z is the impedance matrix of the surface. It should be

noted that the mobility and impedance matrices in Eqs. (3) and

(4), for both the interface and connected acoustic domains, are

independent of external excitation force Fext and thus consid-

ered as the inherent properties of the subsystems. These proper-

ties are determined individually before the assembly.

Consider now the coupling between the two interfaces

through the ventilation duct in Fig. 1. The coupled equations

can be derived by extending the generalized Eq. (3) to incor-

porate a mutual coupling term as

YIðFIext þ ZI

lUI þ ZI

rUI þ ZI–IIUIIÞ ¼ UI;

YIIðFIIext þ ZII

l UII þ ZIIr UII þ ZII–IUIÞ ¼ UII;

(5)

where FIext and FII

ext are the external force vectors applied to

the two interfaces. ZIl and ZI

r are the impedance matrices

describing the acoustic domains on both sides of interface I

(the same applies to interface II). ZI–II is the coupling imped-

ance between the two interfaces, used for describing the

acoustic loading force exerted on interface I due to a vibrat-

ing interface II. Note that the matrix size of ZI–II is N� N0,and ZII–I ¼ ðZI–IIÞT according to the acoustic reciprocity

theory (T stands for transpose operation).

Then, the coupled equations governing the overall sys-

tem response can be summarized into the following form:

TIl Tc

TcT TII

l

" #UI

UII

" #¼ FI

ext

FIIext

" #; (6)

where

TIl ¼ ðYIÞ�1 � ðZI

l þ ZIrÞ;

TIIl ¼ ðYIIÞ�1 � ðZII

l þ ZIIr Þ;

Tc ¼ �ZI–II:

(7)

It can be seen that the transfer functions Tl depend on

the mobility and impedance at the interface locally, whereas

the transfer function Tc describes the mutual interactions

between the two interfaces. Up to this point, the solution pro-

cedure using the proposed sub-structuring approach can be

briefly summarized as follows:

(1) The system ensemble is decoupled into a set of subsys-

tems by their separation interfaces.

(2) A sufficient number of patches are defined for each inter-

face. Then, the local mobility for the interface, the impe-

dances for the acoustic subsystems are determined

(detailed formulations will be demonstrated later).

(3) The general structural-acoustic coupling formulation in

Eq. (3) is employed to perform the system assembling

treatment, and the coupled system response is solved.

The proposed sub-structuring approach provides a sys-

tematic numerical framework rather than a particular tech-

nique to describe the complex vibroacoustic interactions

between subsystems. The subsystem properties can be deter-

mined by employing a broad selection of well-developed nu-

merical methods such as modal approach, FEM, or even a

direct experiment, as long as the transfer function informa-

tion defined at each interface can be obtained. This enables

hybrid models15 (e.g., hybrid modal-FEM16) to be built, if

the subsystems have significantly different orders of n-DOF.

Furthermore, with an aim to conduct system optimization,

the proposed approach can greatly reduce the computational

cost benefiting from the modular treatment.

Then, consider a specific example of a ventilation duct

as shown in Fig. 2. The duct is first assumed to have rigid in-

ternal surfaces and right-angled bends, whose total height is

defined as H1þH2 in Fig. 2, with H2 being the aperture size.

It is noted that although a two-dimensional example is

FIG. 2. A representative ventilation duct with rigid internal surfaces and

right-angled bends.

J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al. 281

presented here in the figure, the following formulation is

generally applicable to three-dimensional cases without any

technical hurdles.

At the duct inlet, the sound field corresponding to the

exterior of the building facade can be described as

Pin ¼ Pi þ Pr; (8)

where Pi and Pr are the incident and reflected sound waves,

respectively. Typically, Pi can be either due to specific

sound sources such as road traffic, or simply assumed as

plane wave incidence for general analysis.

The leakage noise from the outlet domain into the build-

ing interior is

Pout ¼ Pt; (9)

where Pt is the transmitted sound waves, radiated from the

outlet opening.

Using the sub-structuring approach, the system is parti-

tioned into three acoustic subsystems and two interfaces, as

shown in Fig. 3. The two interfaces, 1 and 2, consist of an

aperture and a rigid or flexible structure, which are system-

atically modeled using a proposed Compound Interface

treatment17 by writing their mobilities as

Y1 ¼Ys

Yo

� �; Y2 ¼

Yo

Ys

� �; (10)

where Yo is the equivalent mobility of the aperture by for-

mulating it as a virtual panel;18 Ys is the structural mobility

of the rigid or flexible structure. The detailed formulations

for the aperture and structure have been well-established in

Ref. 18 and are not repeated here.

The rigid ventilation duct with right-angled bends can

be well represented by viewing it as an acoustic cavity.

For illustration purposes, the formulation procedure for

determining its acoustic impedances by using a modal

decomposition approach is presented here. Without loss of

generality, the sound pressure field in a three-dimensional

acoustic cavity can be expanded into the modal coordi-

nates as

pcðx; y; zÞ ¼X

m

amc um

c ; (11)

where pc is the sound pressure at any point inside the cavity;

amc is the mth modal amplitude of the cavity; um

c is the mode

shape function.

For a cavity consisting of a domain Vc and a boundary

Sc, the pressure field inside the cavity can be described

through Green’s function19,20

ðVc

pcr2umc � um

cr2pc

� �dVc

¼ð

sc

pc@um

c

@n� um

c

@pc

@n

� �dSc: (12)

On inserting the Helmholtz equation which governs the

sound wave propagation and making use of the modal ortho-

gonality further gives

amc Nm

c k2 � k2m

� �¼ð

sc

pc@um

c

@n� um

c

@pc

@n

� �dSc; (13)

where km ¼ xm=c0, with xm being the angular eigenfrequen-

cies of the cavity; Nmc ¼

ÐVc

umc um

c dVc, c0 is the speed of

sound in air.

There exist various techniques for solving the above

equation with different approximations.19 A commonly used

one is to assume that the eigenfunctions can be expressed as

a series of rigid-walled acoustic modes

umc ¼ cos

mxpLc

x

x� �

cosmypLc

y

y� �

cosmzpLc

z

z� �

;

mx;my;mz ¼ 0; 1; 2; ::: ; (14)

where mx, my, mz are the modal indices in the x, y, z direc-

tions, and Lcx, Lc

y, Lcz are the cavity dimensions, respectively.

If a vibrating surface defined by Sv presents along the

cavity boundary, the momentum equation is given by

@pc

@n¼ �jq0xU; (15)

where q0 is the air density, and U is the boundary vibrating

velocity. Similarly, an absorbing boundary Sa with its char-

acteristic impedance being specified as Za can be described

as (Neumann condition)20

@pc

@n¼ �jq0x

pc

Za: (16)

Using the cavity mode shape functions in Eq. (14), the

right-hand side term of Eq. (13) in the presence of a vibrat-

ing boundary Sv and an absorbing boundary Sa becomesðsc

pc@um

c

@n� um

c

@pc

@n

� �dSc

¼ð

sv

jq0xUð Þumc dSv þ

ðsa

jq0xam

c

Za

� �um

c umc dSa:

(17)

Consider now the pressure field inside the cavity excited

by a particular vibrating patch q on the cavity boundary.

FIG. 3. Vibroacoustic modeling of the system shown in Fig. 2 using the pro-

posed sub-structuring approach.

282 J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al.

Assuming that a sufficient number of patches have been

defined at interfaces 1 and 2, the individual effect of the

patch with a surface Sq and a vibrating velocity Uq can be

described as

amc Nm

c ðk2 � k2mÞ � jq0x

ðSa

ðumc Þ

2dSa=Za

� �

¼ð

sq

ðjq0xUqÞumc dSq: (18)

Note that subscript q for the exciting patch covers all the

patches discretized along the cavity boundary. Then, by

defining the acoustic impedance as the transfer function

between the force and velocity, exerted onto a receiving

patch p from an exciting patch q, respectively, one has17

Zcpq ¼

Fp

Uq¼

ðsp

pcdSp

Uq¼

ðsp

Xm

amc um

c

� �dSp

Uq: (19)

Inserting Eq. (18) into Eq. (19) gives

Zcpq ¼

Xm

jq0x

Nmc

�k2 � k2

mÞ � jq0xð

Sa

umcð Þ2dSa=Za

�ð

sp

umc dSp

ðsq

umc dSq: (20)

Equation (20) indicates that the impedance expression is

a function of the cavity parameters (Nmc , km, ur

c, Za, Sa) as

well as the local properties at the exciting and receiving

patch (Ð

squm

c dSq andÐ

spum

c dSp). Using the modal expansion

approach, relevant studies have addressed the detailed imple-

mentation procedure for a rigid rectangular cavity,21 and a

general cavity with arbitrary shape and general boundary

conditions.19 The calculated impedances between all the

patches defined on interfaces 1 and 2 are then gathered in

matrices Zc11 and Zc

22 for the local description of the inter-

face, and in matrices Zc12 and Zc

21 for describing their mutual

interactions.

An alternative way to determine these impedance matri-

ces is to use other deterministic models such as FEM. This

can be implemented by calculating the ratio between the

averaged responses at a set of receiving patches, due to the

excitation at one particular patch. This gives one column in

the impedance matrix, like so,

ZN1 ¼ fZ11; Z21; …; ZN1g: (21)

This procedure is then repeated by replacing the excitation

patch by N times to fill in all the elements for ZNN, such that

ZNN ¼ ½ZN1;ZN2; :::;ZNN�.For a rigid rectangular cavity whose eigenfunction is ana-

lytically known in Eq. (14), a simple check has been con-

ducted to compare the patch impedances obtained via using

Eq. (20) and from FEM. Their numerical outputs are basically

identical, which confirms the equivalence of using both meth-

ods to characterize the subsystem property. Nevertheless, the

use of FEM is considered to be more applicable to real win-

dow designs with complex geometries. But the required n-

DOF is also significantly increased compared with the modal

based solution as a major drawback.

In order to predict the ensemble response of the system

as shown in Fig. 3, the coupled equations for describing the

interactions between the two interfaces can be written as

Y1ðF1 þ Zr1U1 þ Zc

11U1 þ Zc12U2Þ ¼ U1;

Y2ðZc21U1 þ Zc

22U2 þ Zr2U2Þ ¼ U2; (22)

where Y1 and Y2 are the mobilities of the two interfaces as

depicted in Eq. (10); Zr1 and Zr

2 are the radiation impedances

of the inlet and outlet acoustic domains, which can be of var-

ious types. F1 is the excitation force being applied to the

inlet, and U1 and U2 are the vibrational responses of the two

interfaces, respectively.

Once the velocity vectors from the coupled equations in

Eq. (22) are solved, the silencing performance of the ventila-

tion duct can be evaluated by calculating its sound transmis-

sion loss (TL). Here, two different TL definitions are

suggested as

TL1 ¼ 10log10ðPa1=P

a2Þ;

TL2 ¼ 10log10ðPaþs1 =Pa

2Þ;(23)

where Pa1 ¼ jp0j2Sa

1=ð2q0c0Þ is the input sound power at the

inlet aperture, if normal plane waves with a pressure ampli-

tude p0 is applied. Pa2 ¼

ÐSa

2

12

p2 � U�2ð ÞdSa2 is the transmitted

sound power from the outlet aperture toward the down-

stream, where p2 is the averaged radiation patch pressure

and U�2 is the complex conjugate of the averaged patch ve-

locity. Sa1 and Sa

2 are the surface areas of the inlet and outlet

apertures, respectively. The difference between TL1 and TL2

simply consists in whether the incident power is calculated

with reference to the inlet aperture only, or the compound

surface inclusive of the structural part (total area of interface

1, Saþs1 ). Given a normal plane wave incidence with a con-

stant amplitude, TL1 can be converted into TL2 using a sim-

ple formula, if the structural part is deemed as rigid5

TL2 ¼ 10 log10

Sa þ Ss

Sa10TL1=10

� �

¼ TL1 þ 10 log10

Sa þ Ss

Sa

� �; (24)

where Sa and Ss are the surface areas occupied by the aper-

ture and structure, respectively. Therefore, in the following

numerical examples, only TL1 is presented without any spe-

cial notice.

III. NUMERICAL EXAMPLES

A. Empty ventilation duct

Consider the two-dimensional ducted ventilation system

as shown in Fig. 2. Sections III A–III D present numerical

studies for evaluating its acoustic performance using the pro-

posed sub-structuring approach. As sketched in Fig. 4, the

J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al. 283

dimension of the ventilation duct is chosen as H1¼ 0.7 m,

H2¼ 0.3 m, and W¼ 0.3 m, where two semi-infinite wave-

guides are connected as the inlet and outlet acoustic

domains. The sound incidence to the inlet aperture is

assumed as a normal plane wave with constant pressure am-

plitude across the frequency range. In such case, the sound

energy is directed into and out of the duct without any loss

elsewhere. Here, the expressions of Zr1 and Zr

2 in Eq. (22),

corresponding to semi-infinite ducts, are derived analytically

based on the modal expansion approach as given in earlier

publications.17,22

For this empty ventilation duct with rigid internal surfa-

ces, validation is first carried out by comparing the system

TL predicted using the proposed approach with that of a

FEM simulation. This also benchmarks the silencing per-

formance of the empty duct. In order to reach a frequency

range up to 4000 Hz, the patch size defined over the coupling

interface is required to be smaller than Dpatch ¼ ks=2 ¼ 340=ð2� 4000Þ � 0:04 m. Therefore, both interfaces 1 and 2 in

Fig. 3 (total height¼ 1 m) are segmented into 30 patches,

with 9 patches over the aperture and 21 patches on the struc-

ture. The structural part is assumed as rigid here with a null

mobility.

A FEM model using commercial software COMSOL is

also built for validation, where the inlet is excited by a plane

wave radiation and the outlet end has a perfectly matched

layer.23 The enclosed acoustic domain is discretized into

40 000 nodes with a tightened criterion of Dfem ¼ ks=8

� 0:01 m. In Fig. 4, the TL results of the empty bended duct

predicted by using the proposed approach and FEM simula-

tion are presented from 10 to 4000 Hz, with a frequency step-

size of 15 Hz. The excellent agreement between the two

curves can be clearly seen, which validates the prediction ac-

curacy of the proposed approach. However, in terms of com-

putational time, the FEM model took around 300 s (5 mins) to

run a simulation while the proposed approach requires only

12 s for the same accuracy (both on a laptop with Intel i5

processor, 4 GB memory). The reason is that the proposed

approach built on a modal basis greatly reduces the n-DOF

compared to that of the FEM model with a nodal basis, which

would become even more crucial if a three-dimensional or

larger system size is to be analyzed. It is worth noting that

such significant reduction also benefits from the simple sub-

system geometry (rectangular cavity and duct cross-sections),

as formulated in Sec. II. For complex geometries requiring the

use of FEM to obtain the subsystem characteristics, the calcu-

lation efficiency will be compromised to a certain degree.

It is also seen from Fig. 4 that, although the TL peaks of

the empty duct can reach up to 25 dB at some particular fre-

quencies, the averaged sound attenuation is still rather low.

The TL lower-bound is typically capped at 5 dB across the

frequency range, with a strong resonant pattern being

observed. By separating the sound power being reflected

back to the inlet and that being absorbed inside the ventila-

tion duct, the system reflection coefficient R and absorption

coefficient a can be calculated as24

R ¼ Pr1=P

a1 ¼ ðPa

1 �Pt1Þ=Pa

1;

a ¼ ðPt1 �Pa

2Þ=Pa1;

(25)

where Pr1 and Pt

1 are the reflected and transmitted sound

powers at the inlet aperture, respectively. As shown in Fig. 5,

the empty duct exhibits only the reactive behavior for sound

attenuation with zero absorption, which is as expected.

Therefore, one possible solution for enhancing the sound

attenuation is to consider the application of sound absorbing

treatment inside the empty ventilation duct, in order toFIG. 5. Reflection coefficient R and absorption coefficient a of the empty

ventilation duct.

FIG. 6. (Color online) (a) Ventilation duct with integral MPP absorber for

providing internal sound absorption. (b) Modified vibroacoustic formulation

at interface 2.

FIG. 4. Comparison of the TL results from the proposed approach and FEM

for an empty ventilation duct. The system dimension is as sketched.

284 J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al.

compensate for the deficiencies at numerous frequencies

observed in Fig. 4.

B. Effect of adding MPP absorber

Traditional sound absorbing materials such as fiberglass

or mineral wool can be well described using Eq. (16)

through the complex acoustic impedance Za. However, fi-

brous absorbing materials have the disadvantages of being

flammable, bulky, non-durable, and unsafe for human health.

Instead, a MPP absorber25 provides a non-fibrous alternative,

and its effectiveness is discussed here. Consider the ventila-

tion duct with an integral MPP absorber as shown in Fig.

6(a). The MPP surface is placed directly facing the inlet

aperture, forming a backing acoustic cavity of depth D with

the rear wall. The duct width left for air passage is decreased

from W to W - D accordingly.

Previous studies17,24 have emphasized the importance of

modeling MPP as an integral part of the vibroacoustic sys-

tem, since its in situ sound absorption is strongly dependent

on the surrounding environment. Comparing the system

shown in Fig. 6(a) with the empty duct in Fig. 2, the cou-

pling through the inlet interface remains unchanged.

Therefore, only the modeling for interface 2 is considered in

Fig. 6(b). For the modeling of the MPP surface, an equiva-

lent mobility treatment has been developed, which averages

the vibration of the panel frame and air-mass motion inside

the micro-holes and gives24

Ympp ¼ 1� rð ÞYp þ rZhDs

� �I; (26)

where Yp is the structural mobility of the panel; r is the per-

centage of area occupied by the holes, or referred to as the

perforation ratio; Ds is the surface area of the divided patch;

I is an identity matrix of the same order as the number of

patches; Zh is the characteristic acoustic impedance of a sin-

gle hole, with its resistance part Zrh and reactance part Zi

h

being expressed as

Zrh ¼

32gt

d21þ k2

32

� �1=2

þffiffiffi2p

32k

d

t

" #;

Zih ¼ jq0xt 1þ 9þ k2

2

� ��1=2

þ 0:85d

t

" #; (27)

where g is the air viscosity, t is the MPP thickness, d is the

hole diameter, and k is the perforation constant

k ¼ dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq0x=4g

p. Note that Eq. (26) is generally applicable

to both rigid and flexible MPPs.

Using the same compound interface treatment as in Eq.

(10), the mobility of interface 2 can be determined by com-

bining the aperture mobility and equivalent MPP mobility as

Y2 ¼Yo

2

Ympp2

� �: (28)

As to the modeling of the radiation domain, this can be

done by slightly changing Zr2 in Eq. (22) to

Zr2 ¼

Zo2

Zbc2

� �; (29)

where Zo2 is the radiation impedance at the upper opening of

interface 2. Duct impedance is used here in the simulation

examples. Zbc2 is the impedance of the acoustic cavity back-

ing the MPP. The same formulation as in Eq. (20) is

applicable.

The sound absorption performance of the MPP absorber

strongly depends on a set of parameters including the hole

diameter d, panel thickness t, perforation ratio r, backing

cavity depth D, as well as the dimension and aperture size of

the duct. In the simulation, the MPP parameters are first

assumed as: d¼ t¼ 0.2 mm, r ¼ 1%, D¼ 0.1 m, with the

same duct dimension as in Fig. 4. In Fig. 7, the predicted TL

of the system with the MPP absorber is compared with the

benchmark TL of the empty duct. It can be seen that the

added MPP absorber significantly enhances the sound

attenuation of the empty duct, resulting in an averaged 5 dB

increase across the frequency range and a 10 dB increase in

some particular frequencies. This clearly demonstrates the

potential of using MPP to achieve a better sound insulation.

In order to quantify the actual sound absorption provided

by the MPP absorber, the reflection and absorption coeffi-

cients of the system are calculated and plotted in Fig. 8. It

can be observed that the sound absorption is prominent and

becomes the dominant mechanism for sound attenuation,

while the reflection effect seems to provide compensations at

the absorption valleys. To explain this, the sound waves

entering into the duct through the inlet aperture first interact

with the MPP absorber before the reactive effect takes place.

At the absorber’s effective frequencies, the sound energy is

efficiently damped by the distributed tiny holes and radiation

back to the main duct is reduced, as evidenced by the low

value in the reflection coefficient R. When the MPP absorp-

tion is less efficient due to resonances of the backing cavity

(a valleys), the bended duct as a reactive device still enables

strong sound reflection as explained in Fig. 5. Such a hybrid

sound attenuation mechanism ultimately maintains the sys-

tem TL at a relatively high level as seen from Fig. 7.

FIG. 7. Effect of the additional MPP absorber on TL.

J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al. 285

It is also interesting to note that the actual sound absorp-

tion curve of the MPP closely resembles the theoretical

curve when it is placed in an impedance tube for testing

(known as Maa’s theory25). This observation also supports

the above discussion and indicates that the absorption mech-

anism is primary and dominating. Based on this, one may

suggest that a better absorption design can be achieved by

selecting the right MPP absorber with optimized parameters.

But still, the overall silencing performance is determined by

the hybrid effects of sound reflection and absorption. Aiming

an optimal TL response, one may need to balance these two

effects for a desired broadband performance in the design

frequency of interest.

C. Parametric studies

Following the previous mechanism investigations, a nu-

merical study for analyzing the possible influence of system

parameters is presented. It is assumed that the basic system

configuration in Fig. 6 remains unchanged for all the simu-

lated cases, whereas the parameters associated with the MPP

absorber can vary within a certain range. In order to alter the

impedance of the MPP surface, the hole diameter d and its

influence are first considered. In Fig. 9, the TL responses

corresponding to three MPP cases with the hole diameter dvarying from 0.2 to 1 mm are compared. The MPP thickness

t is simply assumed as equal to d, and other parameters are

fixed as: r ¼ 1%, D¼ 0.1 m. It can be seen that a smaller

hole diameter generally leads to a better silencing perform-

ance in a medium frequency range from 400 to 2000 Hz. Its

correlation with the absorption coefficient curve a has been

checked, which confirms that the improvement is mainly

attributed to an enhanced and widened absorption band pro-

vided by the MPP with smaller holes. It is then suggested

that an efficient sound absorption would call for a small hole

size d (the panel thickness t needs to be matched), in agree-

ment with similar observations reported in some other stud-

ies.24–26 However, it is worth mentioning that this argument

is not universally true and requires other factors such as

backing cavity size and intended frequency range to be con-

sidered simultaneously. Beside the hole diameter, similar

analyses on the panel thickness and perforation ratio have

also been performed (not shown here). The major conclusion

from these analyses is that the system TL response is very

sensitive to the variation of MPP parameters.

By fixing the perforation parameters as d¼ t¼ 0.2 mm,

r ¼ 1%, which gives the best broadband performance in

Fig. 9, the influence of backing cavity size of the MPP is

investigated by comparing three cases with D¼ 0.05, 0.1,

and 0.15 m, respectively. As shown in Fig. 10, the variation

of the TL curves across the frequency range of interest is

clearly observed. But unlike the previous discussion on the

hole diameter, no general variation trend can be concluded

from the comparison in Fig. 10. The reason is that different

cavity depths not only affect the absorption curve a by alter-

ing the resonant behavior of the absorber, but also changes

the reflection curve R due to the fully coupled nature of the

acoustic fields.

From the above analyses, it can be seen that the parame-

ter changes provide considerable scope for the tuning of sys-

tem TL, whilst the proposed sub-structuring approach offers

a computationally tractable tool. Before starting the optimi-

zation study, some interim conclusions rising from the pres-

ent analyses can be summarized:

FIG. 8. (Color online) Reflection coefficient R and absorption coefficient aof the ventilation duct with a MPP absorber. The dotted line plots he Maa’s

theory for a MPP absorber in impedance tube setting.

FIG. 9. TL variations with varying MPP hole diameter d.

FIG. 10. TL variations with varying backing cavity depth D of the MPP.

286 J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al.

(1) In order to achieve a better sound attenuation, control-

ling the MPP perforation parameters seems to be more

important than varying the backing cavity size. The for-

mer determines the absorption ability and bandwidth by

changing the surface impedance, while the latter only

alters the resonant behavior of the coupled acoustic

fields.

(2) Generally, a smaller hole diameter d can offer a more

effective sound dissipation and a broader absorption

bandwidth. An appropriate panel thickness t and perfora-

tion ratio r also needs to be matched with the selection

of d.

(3) The influences of the involved parameters are intercon-

nected. In order to seek the best result, one needs to per-

form system optimization with multiple variables.

(4) A balance between the sound reflection and absorption

effect is needed for achieving an overall broadband TL

performance.

D. Optimization study

The proposed sub-structuring model significantly

reduces the computational cost compared with FEM and is

employed to perform the system optimization. By assuming

a fixed duct dimension and aperture size as in Fig. 4, the pa-

rameters associated with the MPP absorber ft; d; r;Dg are to

be optimized for a given frequency range of interest. The

objective function is defined as to minimize the total sound

power transmitted from the outlet aperture, which integrates

the value of Pa2 at all the discrete frequency points.27 This

enables us to identify the best combination of parameters

which gives the best broadband TL performance. Being

mindful of the validity of Eq. (27), the hole diameter d (as

well as panel thickness t) is varied from 0.2 to 1 mm. The

perforation ratio r is constrained within 0.3% to 2%, and the

backing cavity depth D can vary from 0.02 to 0.15 m. Note

that the flow rate consideration is temporarily neglected,

which can be integrated into the optimization model for a

more practical analysis.

The first optimization study (referred to as Opt 1) con-

siders d and D as variables, where the perforation ratio r is

fixed at 1%. The targeted frequency from 400 to 2000 Hz is

linearly divided into 110 frequency points for the integration

of the total power P. The d interval (0.2–1 mm) is divided

into 17 points with an increment of 0.05 mm, and the Dinterval (0.02–0.15) m is divided into 27 points with an in-

crement of 0.005 m. This results in 459 combinations for the

exhaustive search of the optimal TL. Using the proposed

approach, the calculation time required for each case is

roughly 9 s. The total time for the optimization is therefore

459� 9 � 4000 s (about 1 h).

In Fig. 11, the distribution of the calculated total power

as a function of the two variables d and D is presented,

which demonstrates the existence of the minima at

d¼ t¼ 0.2 mm, D¼ 0.06 m (total power P¼ 0.0037 w). By

looking at a particular value of D while gradually decreasing

the hole diameter d, the general trend of an increased sound

attenuation performance is observed. However, by fixing dat a certain value while varying the cavity depth D does not

show any generalized and conclusive behavior. Both obser-

vations are in consistent with the observations made in Sec.

III C.

Then, consider a second optimization study (Opt 2) with

the inclusion of an additional variable, perforation ratio r.

The frequency and D intervals are the same as Opt 1,

whereas the d interval is narrowed to (0.2–0.5) mm with 7

points. The r interval (0.3%–2%) is evenly divided into 18

points with an increment of 0.1%. This results in a total

number of 7� 27� 18 � 3400 cases, and the total computa-

tional time is extended to 8.5 h. The minima of the total

sound power from Opt 2 is located at d¼ t¼ 0.2 mm,

D¼ 0.05 m, r¼ 0.5%, and the total power P of this combi-

nation is further decreased to 0.0029 w, compared with

0.0037 w in Opt 1. This leads to an average 1 dB improve-

ment in the overall sound reduction curve.

In Fig. 12, the TL curves corresponding to the optimal

results from Opt 1 and 2 are presented. Both curves are seen

to exhibit broadband attenuation performance in the targeted

frequency range, with an averaged TL of about 15 dB. The

little improvement in the TL response from Opt 2 can be

also identified by looking at the TL lower-bound. Further

FIG. 11. (Color online) Total power vs MPP hole diameter d and backing

cavity depth D, for the first optimization case study (Opt 1).

FIG. 12. Optimized TLs from the two optimization case studies.

J. Acoust. Soc. Am. 139 (1), January 2016 Yu et al. 287

analysis on the reflection and absorption coefficients pro-

vides insight to the optimized results. In Fig. 13, the R and acurves corresponding to the optimized result from Opt 2 are

plotted. Below 1600 Hz, the sound absorption provided by

the MPP absorber is more effective than the sound reflection

provided by the bended duct. After the a maximum at

800 Hz, the gradual decrease in the absorption curve is com-

pensated by an increase in the reflection curve. The two

curves intersect at around 1600 Hz, where R¼ a¼ 0.5. This

eventually shows the well-balanced behavior of the hybrid

effects in the optimized configuration.

IV. CONCLUSIONS

A sub-structuring approach to model the complex

structural-acoustic coupling involved in ducted ventilation

systems has been proposed. The system ensemble is

decoupled into a set of subsystems, allowing the characteri-

zation of their transfer functions at the separation interfaces.

The assembling treatment is performed by using a patch

based interface matching technique. For a representative

ventilation duct with regular geometry, a numerical model

has been built to predict its sound attenuation performance.

The involved subsystems were modeled analytically using

modal based solutions. By comparing the simulation results

with a FEM model, the convergence and prediction accuracy

of the proposed model have been validated.

Through the presented numerical examples, the poten-

tial of using a MPP absorber to improve the sound attenua-

tion performance of an empty ventilation duct has been

shown. With MPP, the system exhibits a hybrid sound

attenuation mechanism with combined reactive and dissipa-

tive effects. The sound absorption provided by the MPP

absorber basically dominates the overall attenuation, and the

absorption valleys due to the backing cavity resonances are

compensated by the sound reflection provided by the bended

duct.

The sensitivity of the system response to parameter

changes has been shown. In particular, the effects of varying

MPP hole diameter d and backing cavity depth D were

discussed. These two parameters are effective in controlling

the MPP surface impedance and the resonant frequencies of

the system. Using the proposed approach, two optimization

studies were conducted to search for the optimal combina-

tion of parameters. The optimized results exhibited broad-

band attenuation performance through well-balancing of the

sound reflection and absorption effects. The flow rate analy-

sis can be further included to guide a more comprehensive

design and optimization for the present device.

As a general comment on the merits of the proposed

approach, the associated computational cost was shown to

be significantly lower compared with FEM. Plus with the

modular nature of the approach, optimization studies were

able to be carried out efficiently. Meanwhile, the proposed

sub-structuring modeling framework enables the construc-

tion of hybrid numerical models, where the subsystem prop-

erties can be obtained using different numerical methods.

Benefiting from a patch based interface treatment, system as-

sembly at the connecting boundaries can be performed in a

very convenient and flexible way. This could further extend

the applicability of the proposed approach to tackle more

challenging engineering applications involving subsystems

with complicated geometries or different orders of n-DOF,

for fulfilling the real design need of such systems.

ACKNOWLEDGMENT

This material is based on research/work supported by

the Singapore Ministry of National Development and

National Research Foundation under L2 NIC Award No.

L2NICCFP1-2013-9. The authors also wish to acknowledge

two grants from the Research Grants Council of Hong Kong

Special Administrative Region, China (Grant Nos. PolyU

5103/13E and PolyU 152026/14E).

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